If you read the docs, you'll find that you shouldn't be using subroutine references. Instead, you should be using strings as your method names.

package FlierReports;

use strict;
use Data::Dumper;

use base qw/ FlierFunctions /;

sub setup {

        my $self = shift;

        # Setup the run mode stuff
        $self->start_mode( 'openOrders' );
        $self->mode_param( 'rm' );
        $self->run_modes(
                openOrders      => 'openOrderSummary',
                detail          => 'openOrderDetail',
                markDone        => 'markFinished',

        );

}
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Do that in every class and everything will "just work".